v1.5.0 — Villarceau Ribbons

Villarceau Ribbons joins the catalogue as the 36th pattern — and it is the first here whose motion closes by a group law rather than by a construction we impose.

The 3-sphere is a disjoint union of great circles, one per point of an ordinary sphere. Stereographic projection sends each of those fibers to an exact circle, and the fibers over a latitude form a torus of mutually linked circles — the Villarceau circles the pattern is named for. Motion is left multiplication by a one-parameter SO(4) subgroup; at a full turn that element is −1, which lies inside every fiber's own circle group, so every fiber returns to itself as a set. The loop is the group law, not a trick.

All 84 fibers move across a cycle, median travel 96px — no frozen elements — at p50 2.4ms.

Also in this release

• A stage-fitness audit of all 36 patterns, measuring what actually makes a pattern work as a music visualizer: smooth structural params, chaotic ones, intrinsic time, visible motion fraction, silence behaviour, preset reach.
• Hitomezashi gains its first continuous structural axis. The audit measured it with none — audio could reach stroke weight and scale and nothing else, which is why it barely responded. Its stitch inset is now interpolated; at the default the expression is exactly the historical value, so every existing URL renders byte-identically.
• Seven mis-routed presets fixed. Interlace, isoweave, linefield, knot, chirp and node garden each had a structural parameter sitting unused while a preset slot went to a second cosmetic route.
• Four candidate params were measured and rejected rather than routed — where a preset already routed 100% structural, or where an established NEVER_ROUTE finding outranked a metric known to be blind to that case.

Full history: v1.4.0...v1.5.0